Final answer:
To find the stationary points of the curve y=x³-6x², we first determine the first derivative and set it to zero, finding that x=0 and x=4. Substituting these into the original function, we get stationary points (0,0) and (4,-32). Using the second derivative, we determine that (0,0) is a relative maximum and (4,-32) is a relative minimum.
Step-by-step explanation:
To find the coordinates of the stationary points of the curve y= x³- 6x², we first need to find the derivative of the function, which represents the slope of the tangent to the curve at any point. If the slope is zero, we have a stationary point.
Finding the First Derivative
The first derivative of the function is:
y' = 3x² - 12x
To find the stationary points, set the first derivative to zero:
3x² - 12x = 0
x(3x - 12) = 0
Therefore, x = 0 or x = 4.
Stationary Points
Substitute these values into the original function to find the y-coordinates:
For x = 0, y = 0
For x = 4, y = 4³ - 6(4²) = 64 - 96 = -32
The stationary points are (0,0) and (4,-32).
Determining Maximum or Minimum
To determine whether these points are maxima or minima, we take the second derivative:
y'' = 6x - 12
For x = 0: y'' = 6(0) - 12 = -12, which is negative, indicating a relative maximum.
For x = 4: y'' = 6(4) - 12 = 24 - 12= 12, which is positive, indicating a relative minimum.